# Maths

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• #69254
Harry
Keymaster

Got any questions about maths? You have a huge community of experts on this forum, so why not ask them here. No coursework help though!

#89060
Participant

To kick-start this topic:

MAGIC SQUARES

Definition: A magic square is defined to be a square of numbers such that each row, column and (major) diagonal has the same sum. As an example,
2 9 4
7 5 3
6 1 8
has a sum of 15 along each row/column/diagonal.

Problems: Complete the following magic squares.

#1
02 12 10
?? 08 ??
?? ?? ??

#2
13 20 ??
?? 14 ??
?? ?? ??

#3
Show that if a, b and c are known below then there is a unique solution:
a b ?
? c ?
? ? ?

#89094
Participant

MAGIC SQUARES – PART 2

(P.S. Feel free to extend/reply as you wish!)

#4
Show that the following square has no solutions.
1 2 3
? 4 ?
? ? ?

#5
Find x below.
12 ?? ??
?? x ??
?? ?? 02

#6
Complete this magic square.
15 01 ??
?? ?? 13
?? ?? ??

#89428
AB1
Participant

1:02,12,10
16,08,00
06,04,14

2:13,20,09
10,14,18
19,08,15

#89429
AB1
Participant

4: [possible numbers are 1,2,3,4,5,6,7,8,9]
top row: 1+2+3 = 6
using this, this would mean that 1+4+bottom right square should be 6
however, the only possibility the the bottom right square is 1, which is already used
therefore, the square has no solutions

5: x=7

6:15,01,11
05,09,13
07,17,03

#89500
Rhydwen23
Participant

The problem of constructing magic squares with prime numbers only was first discussed by Henry E. Dudeney in The Weekly Dispatch for 22nd July and 5th August 1900. Dudeney went on to describe a 3rd order magic square comprised of nine different prime numbers* with the lowest possible constants.

Can you find it?

*Dudeney included the number 1 as a prime number for this exercise.

#89569
Participant

@AB1, nice solutions. For #4, it turns out that even if repeat numbers were valid, there still wouldnâ€™t be a solution. Can you find out why?

@Rhydwen23, I have that any magic square of primes can only consist of numbers that are either all 1 more than a multiple of 6 or all 1 less than a multiple of 6. Because of this I claim that the smallest solution that satisfies your conditions is:
31 73 7

#89571
Participant

[Sequel to post #89569]
Sorry! Accidentally pressed the â€˜Submitâ€™ buttonâ€¦(!!!) The solution I had is:
31 73 07
13 37 61
67 01 43
(or equivalent)

#89661
Participant

MAGIC SQUARES – PART 3

#7
Show that no 2×2 magic squares exist (other than the trivial case where all numbers in the square are equal).

#8
Solve the following 4×4 magic square (repeat numbers valid) – decimals may appear!
?? ?? 20 02
?? ?? ?? 01
22 ?? ?? ??
05 10 20 23

This is a bit more relevant to the National Cipher Challenge: 2002 marked the year of the (01)st Cipher Challenge. The 22nd edition will commence on 05/10/2023.

#89669
Rhydwen23
Participant

The solution, for this 3rd order, and up to 12th order, magic square of prime numbers is provided in an article in the Monist (Chicago) for October 1913 by W S Andrews and H A Sayles. This is available on-line.

For anyone not wishing to include the number 1 as a prime, then I think the 3rd order magic square of prime numbers, with the lowest factor (177) is:

101 005 071
029 059 089
047 113 017

#91126
The_Letter_Wriggler
Participant

INFORMATIONAL POST – MAGIC SQUARES
===================================
THE WONDER OF THE NUMBER 1480028

USING THE NUMBERS: 129 141 153 159 171 183 189 201 213
WE FORM A 3 DIGIT MAGIC SQUARE WITH CONSTANT 513

141 189 183

213 171 129

159 153 201

We find that if we put these numbers onto the end of 1480028 they make nine primes:
1480028129 1480028141 1480028153 1480028159 1480028171 1480028183 1480028189 1480028201 1480028213

AND SO WE CAN FORM A 10 DIGIT ALL PRIME MAGIC SQUARE WITH CONSTANT 4440084513

1480028141 1480028189 1480028183

1480028213 1480028171 1480028129

1480028159 1480028153 1480028201

#91127
The_Letter_Wriggler
Participant

TLW 2 Quickies
==============

===================================================
What number gives the same result
when it is added to 1/2
as when it is multiplied by 1/2?

Q2: The devil does math
========================
He takes HIS number and squares it
He cubes each digit and then adds them all together (answer 1)

He takes HIS number and cubes it

He smiles to himself, why?
Bonus: Can you write it out as a math statement?

#91447
Rhydwen23
Participant

The Devil may have all the best tunes, but he doesn’t have a monopoly on numbers of the type described by The_Letter_Wriggler in
#91127 Q2: The devil does math. Can you find another number that matches the criterion laid out in Q2, other than the Devil’s number itself?

[I haven’t had time to check your calculations, but then I don’t need to.! With such a talented crew taking part, someone should be able to come up with an answer! Harry]

#91479
The_Letter_Wriggler
Participant

@Rhydwen23
Never though to test for that! But yes a four digit number N
(so as not to give it away) whose 12 divisors sum to 4368

#91638
Rhydwen23
Participant

Is this right?

The Prime Counting Function (PCF) says the number of primes less than or equal to N is given by N/ln(N), where ln(N) is the natural logarithm of N. For example the Prime Counting Function suggests 21.715 primes less than or equal to 100; suggesting a probability of any single integer between 1 and 100 being prime is 21.715 / 100 = 0.21715.

For TLW’s ten-digit magic square the smallest and largest numbers are:

N = 1480028213, PCF = 70092602.11
N = 1480028129, PCF = 70092598.32

A range of 84.

It seems risky to subtract two large numbers and put any faith in the small remainder, but ignoring that, the PCF suggest just 3.79 prime numbers lie between 1480028129 and 1480028213. Giving a probability of any particular single number being prime as 3.79 / 84 = 0.04512. The probability of all, of any nine numbers from within that range, being prime is (3.79/84)^9 = 7.75 x 10^-13. That seems a vanishingly small probability and suggests a remarkable result to find the case that beats the odds.

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